Method and arrangement for medical x-ray imaging and reconstruction from sparse data

ABSTRACT

The invention relates to a medical X-ray device  5  arrangement for producing three-dimensional information of an object  4  in a medical X-ray imaging medical X-ray device arrangement comprising an X-ray source  2  for X-radiating the object from different directions and a detector  6  for detecting the X-radiation to form projection data of the object  4.  The medical X-ray device  5  arrangement comprises: 
         means  15  for modelling the object 4 mathematically independently of X-ray imaging    and means  15  for utilizing said projection data and said mathematical modelling of the object in Bayesian inversion based on Bayes&#39; formula  
         p   ⁢     (     ⁢   x   ⁢        m   )       =             p   pr     ⁡     (   x   )       ⁢     p   (   m        ⁢   x     )       p   ⁡     (   m   )             
 
to produce three-dimensional information of the object, the prior distribution p pr (x) representing mathematical modelling of the object, the object image vector x, which comprise values of the X-ray attenuation coefficient inside the object, m representing projection data, the likelihood distribution p(m|x) representing the X-radiation attenuation model between the object image vector x and projection data m, p(m) being a normalization constant and the posteriori distribution p(x|m) representing the three-dimensional information of the object  4.

BACKGROUND OF THE INVENTION

Three-dimensional X-ray Imaging is based on taking several 1-D or 2-Dprojection images of a 3-D body from different directions. If 1-Dprojection images are available from all around a 2-D slice of the bodywith dense angular sampling, the inner structure of the slice can bedetermined. This Is known as Computerized Tomography (CT) imagingtechnology, which is widely used in medicine today. A crucial part of CTtechnology is the reconstruction algorithm taking the X-ray images asargument and producing a voxel representation of the 3-D body.

In many practical cases X-ray projection Images are available only froma limited angle of view. A collection of X-ray images of a 3-D body iscalled sparse projection data if (a) the images are taken from a limitedangle of view or (b) there are only a small number of Images. Sparseprojection data does not contain sufficient information to completelydescribe the 3-D body.

However, some a priori information about the body is typically availablewithout X-ray imaging. Combining this information with sparse projectiondata enables more reliable 3-D reconstruction than is possible by usingonly the projection data.

Traditional reconstruction algorithms such as filtered backprojection(FBP), Fourier reconstruction (FR) or algebraic reconstruction technique(ART) do not give satisfactory reconstructions from sparse projectiondata. Reasons for this include requirement for dense full-angle samplingof data, difficulty to use a priori information for examplenonnegativity of the X-ray attenuation coefficient and poor robustnessagainst measurement noise. For example the FBP method relies on summingup noise elements with fine sampling, leading to unnecessarily highradiation dose.

BRIEF DESCRIPTION OF THE INVENTION

The aim of the invention is to overcome the problems met in 3-Dreconstruction of the body that occur when using traditionalreconstruction algorithms with sparse projection data. This is achievedby a method for producing three-dimensional information of an object inmedical X-ray imaging in which method the object is modelledmathematically Independently of X-ray imaging. The object is X-radiatedfrom at least two different directions and the said X-radiation isdetected to form projection data of the object. Said projection data andsaid mathematical modelling of the object are utilized in Bayesianinversion based on Bayes' formula${p\left( x \middle| m \right)} = \frac{{p_{pr}(x)}{p\left( m \middle| x \right)}}{p(m)}$to produce three-dimensional information of the object, the priordistribution p_(pr)(x) representing mathematical modelling of theobject, x representing the object image vector, which comprises valuesof the X-ray attenuation coefficient inside the object, m representingprojection data, the likelihood distribution p(m|x) representing theX-radiation attenuation model between the object image vector x andprojection data m, p(m) being a normalization constant and theposteriori distribution p(x|m) representing the three-dimensionalinformation of the object.

The invention also relates to a medical X-ray device arrangement forproducing three-dimensional information of an object In a medical X-rayimaging, said medical X-ray device arrangement comprises:

-   -   means for modelling the object mathematically independently of        X-ray imaging    -   an X-ray source for X-radiating the object from at least two        different directions,    -   a detector for detecting the X-radiation to form projection data        of the object,    -   and means for utilizing said projection data and said        mathematical modelling of the object in Bayesian inversion based        on Bayes' formula        ${p\left( x \middle| m \right)} = \frac{{p_{pr}(x)}{p\left( m \middle| x \right)}}{p(m)}$        to produce three-dimensional information of the object, the        prior distribution p_(pr)(x) representing mathematical modelling        of the object, x representing the object Image vector, which        comprises values of the X-ray attenuation coefficient inside the        object, m representing projection data, the likelihood        distribution p(m|x) representing the X-radiation attenuation        model between the object Image vector x and projection data m,        p(m) being a normalization constant and the posteriori        distribution p(x|m) representing the three-dimensional        information of the object.

The invention is based on that biological issues have that kind ofstatistical a priori information that this information can be utilizedsuccesfully with bayesian inversion in medical x-ray imaging. Thesuitable a priori information makes possible to model the biologicaltissue mathematically accurately enough and independently of X-rayimaging. From biological tissue it is possible to compile kvalitativestructural information which makes it possible to utilize the bayesianmethod successfully to solve the problems in medical three-dimensionalx-ray imaging. There are certain regularity in biological tissues andthis regularity is useful especially with the bayesian method.

For example 10 x-ray images is taken from breasts of different persons.From these x-ray Images is noticed that there is much similarity in thestatistical structure of the breasts between different people. In otherwords biological tissues and x-ray images taken from the biologicaltissues has similar or almost similar statistical structure betweendifferent persons.

With Bayesian inversion it is possible to utilize a priori informationefficiently in 3-D reconstruction from sparse projection data, becausethe suitable a priori information from the biological tissue makespossible to model the biological tissue mathematically accurately enoughand independently of X-ray imaging. Any collection of projection datacan be used in the reconstruction. Application-dependent a prioriknowledge can be used to regularize the ill-posed reconstructionproblem.

This invention improves the quality of 3-D reconstructions overtraditional methods. In addition, the number of radiographs can beminimized without compromising quality of the reconstruction. This isvery important in medical applications since the X-ray dose of thepatient can be lowered.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows a simple example of X-ray imaging.

FIGS. 2A -2B show a difference between global tomography and localtomography.

FIGS. 3A-3B show different types of sparse projection data. Every blackdot represents a location of the X-ray source for taking one projectionimage.

FIGS. 4A-4B show examples of parts of boundaries and cracks of an objectvisible and invisible without a priori information.

FIG. 5 illustrates “pencil beam” X-ray attenuation model.

FIG. 6 is basic flow chart of the method according to the invention.

FIG. 7 is an intraoral X-ray device arrangement presenting one preferredembodiment of the invention.

FIG. 8 shows a measurement geometry of dental limited-angle tomographywith a digital intraoral sensor.

FIG. 9 shows results of 3D reconstruction of head phantom in the firstpreferred embodiment.

DETAILED DESCRIPTION OF THE INVENTION

In practical imaging situations X-ray images are not always availablefrom all around the body. The body might be visible only from certaindirections due to imaging geometry. For example this is the case in 3-Dmammography with the breast compressed against the detector, or inintraoral dental imaging with the detector inside the patient's mouth.This situation is called limited-angle tomography. Also, the region ofinterest might be surrounded by tissue that need not be imaged, like inextraoral dental imaging. This situation is called local tomography. Inaddition the number of radiographs should be minimized in medicalapplications for reducing the X-ray dose of the patient.

In the preferred embodiments of the invention Bayesian inversionalgorithms are used to create a new type of 3-D medical X-ray imaging.It is intermediate between a projection radiograph and a full computedtomography scan. Two steps are needed to perform successfully suchimaging: In step one, the doctor (a) collects all the prior informationhe or she has on the tissue of Interest, and (b) takes the minimumnumber of radiographs containing the desired 3D information. In steptwo, a tomographic algorithm based on Bayesian inversion is used for 3Dreconstruction. The algorithm contains two separate mathematical models.First, all a priori information (i.e. information that is independent ofX-ray imaging) is used to model the unknown object mathematically. Themodel is put into the form of prior distribution in Bayes' formula.Second, the measurement is modelled mathematically. This involves thegeometry of the imaging device, position of the detector and X-raysource during the exposure of each projection image and a model forX-ray attenuation inside tissue. The mathematical model of themeasurement is put into the form of likelihood distribution in Bayes'formula.

In FIG. 1 is shown a simple example of X-ray imaging, where an X-raysource 2 is placed on one side of an object 4 under imaging. Radiationpasses through the object and is detected by a detector 6 on the otherside. The X-ray source is for example a X-ray source of an intraoralX-ray source of a dentist, of a dental panoramic X-ray device, of asurgical C-arm X-ray device, of a mammography device or of any othermedical X-ray device and the detector 6 is a detector of some of thosedevices. Usually the detector 6 is a digital sensor that can be thoughtof as a 2-D array of almost pointlike detectors.

The 3-D body under imaging is modelled by nonnegative X-ray attenuationcoefficient The value gives the relative intensity loss of the X-raytravelling within a small distance dr: $\begin{matrix}{\frac{dI}{I} = {{- {x(r)}}{dr}}} & (1)\end{matrix}$

The X-radiation has Initial intensity I₀ when entering the object 4 anda smaller intensity I₁ when exiting the object. The following equationshows the attenuation law: $\begin{matrix}{{\int_{L}{{x(r)}\quad{\mathbb{d}r}}} = {{- {\int_{L}{\frac{I^{\prime}(r)}{I(r)}\quad{\mathbb{d}r}}}} = {{\log\quad I_{0}} - {\log\quad I_{1}}}}} & (2)\end{matrix}$where initial intensity I₀ is known by calibration, and intensity afterobject I₁ is known from the corresponding point value in a projectionimage. Thus the measured data is the integral of x along the line L.

In the above model it is not taken into account (a) scattering phenomenaresulting in X-rays changing their direction, or (b) the dependency ofattenuation on the X-ray spectrum resulting in low-energy photons beingmore easily attenuated than high-energy ones. Effect (b) causes errorsin measurements and is sometimes referred to as beam hardening. Moredetailed models can be utilized in Bayesian inversion to overcome (a)and (b).

In medical imaging the geometrical arrangements of the X-ray source anddigital sensor vary according to the diagnostic task and equipment.FIGS. 2A-2B and 3 illustrate the types of tomographic data resultingfrom different imaging situations. For clarity, here are presentedtwo-dimensional examples; similar situations can be considered in 3-D.

In FIG. 2A-2B are shown two cases according to whether the whole object4 is fully visible in each projection or not. These cases are calledglobal tomography and local tomography, respectively. In FIG. 2B isradiated only ROI (Region Of Interest) 8.

The types of data described in FIGS. 3A-3B cover a large range ofspecific imaging tasks. The choice of data collection dictates what kindof features and details it is easiest to reconstruct reliably from thedata. Thus the choice of projection views must be made according to thediagnostic task at hand. In traditional CT imaging, projections aretaken from all around the object 4. In the preferred embodiments of theinvention radiation dose is lowered by sampling the angular variablemore sparsely. FIG. 3A presents more dense angular data with full angleand limited angle measurements. FIG. 3B presents more sparse angulardata, also with full and limited angles of measurement.

In FIGS. 4A and 4B there are presented examples of parts of boundary andcracks that are visible or indetectable in reconstruction without apriori information. FIG. 4A there is an object 4 under imaging with edgeon the surface of the object. The edge is detectable on the leftmostposition where the edge is more parallel to the direction of the X-rays.In the FIG. 4B there is a crack inside of the object 4. The crack isdetectable in the leftmost position parallel to the angle ofmeasurement.

The main idea in Bayesian inversion method is to consider the inverseproblem as a problem of statistical inference. All variables areredefined to be random variables. The randomness reflects uncertainty oftheir actual values and the degree of uncertainty is coded in theprobability distributions of these random variables.

When modelling the measurement mathematically the X-ray attenuationmodel and the observation can be assumed linear. The model Is presentedin the following equation.m=Ax+e,   (3)where the variables m, x and e are random variables.

The matrix A comes from the pencil beam model for the X-ray attenuation.This Is presented in FIG. 5. There the unknown 3-D body of the object 4is divided into small voxels 10, and the matrix A contains the lengthsof the path of the X-ray inside each voxel. In this way the integral Informula (2) is approximated with a simple numerical quadrature.

Assume now that the object image vector x and the noise are jointlyindependent random variables. The conditional probability distributionof x given the measurement m is given by Bayes' formula in the form$\begin{matrix}{{{p\left( x \middle| m \right)} = \frac{{p_{pr}(x)}{p_{noise}\left( {m - {Ax}} \right)}}{p(m)}},} & (4)\end{matrix}$where p(m) is a normalization constant The density p(x|m) is called theposteriori distribution of x. p_(noise)(m-Ax)=p(m|x) is a likelihooddistribution representing the X-radiation attenuation model between theobject image vector x and projection data m.

The density called the prior distribution of x, is designed to containall possible information available of the actual object 4 independentlyof X-ray imaging. It is crucial, in contrast to several classicalregularization methods, that the choice of the prior distribution shouldnot be based on the projection data. The rule of thumb in the design ofprior distributions is that typical image vectors (say, of some existinglibrary) should have high prior probability while atypical or impossibleones should have low or negligible probability.

In the framework of the Bayesian Inversion theory, the posteriordistribution in formula (4) represents the complete solution of the 3Dreconstruction problem.

To produce an image of the object 4 based on the posterior distribution,several alternatives exist. The most common ones are the maximum aposteriori estimator (MAP) and conditional mean estimator (CM). They aredefined by the formulasp(xMAP|m)=maxp(x|m),   (5)where the maximum on the right hand side is taken over all x, andxCM=∫xp(x|m)dx.   (6)

Finding the MAP estimator is an optimization problem while finding theCM estimator is a problem of integration.

In FIG. 6 is presented a basic flow chart of the method according to theinvention. In method step 600 the object is modelled mathematicallyindependently of X-ray imaging. In method step 602 the object IsX-radiated from at least two different directions. In method step 604the said X-radiation is detected to form projection data of the object.In method step 606 said projection data and said mathematical modellingof the object are utilized in Bayesian inversion based on Bayes formulato produce three-dimensional information of the object. The method step600 is also possible to perform after method step 602 or 604.

In the first preferred embodiment of the invention is presented anapplication to dental radiology.

X-ray projection images are conventionally used in dental radiology.However, certain diagnostic tasks require more precise knowledge of the3D structure of tissue than is available in two-dimensional radiographs.Such tasks include implant planning and detection of bone loss betweentooth roots.

In FIG. 7 is presented an intraoral X-ray device 5 arrangementpresenting the first preferred embodiment of the invention. It isimportant to notice that this is only an example of the medical X-raydevice 5 arrangement where the invention is possible to be utilized.

The medical x-ray device 5 in the preferred embodiments of the inventionIs for example a dental panoramic X-ray device, a surgical C-arm X-raydevice or a mammography device.

In the first preferred embodiment of the invention the articulated armarrangement 3 moves the X-ray source 2 to the right position. TheX-radiation begins by pressing the exposure button 12. The X-ray source2 X-radiates the object 4, which is for example teeth of a patient Thedetector 6 detects the X-radiation. The Image information which is gotby detecting the X-radiation is sent by communication link 16 to thecomputer 14. The computer comprises the software means 15 to process theimage information according to the invention. There can be more than onecomputer 14 and also the software means 15 can situate in more than onecomputer 14. For example the first computer 14 is computer which is usedin x-ray imaging. The second computer 14 is computer which is used inprocessing the image information according to the Invention. It ispossible to have the second computer 14 far away from the actual medicalx-ray device 5. For simplicity In FIG. 7 is shown only one computer 14.

In the first preferred embodiment (FIG. 7) of the invention, thedentists X-ray equipment is used for taking a set of 2D projectionImages that are used as input for Bayesian 3D reconstruction algorithm.Such equipment includes an intraoral X-ray unit and a digital intraoralsensor.

Benefits of this approach over conventional CT scan are

-   -   low cost and convenient usage,    -   high resolution of the projection images,    -   possibility to take as few radiographs as is needed to capture        the relevant 3D information, minimizing radiation dose to the        patient,    -   the possibility to choose imaging directions so that the X-rays        do not pass through the whole head but only the interesting        tissue, further reducing dose.

In the preferred embodiments of the invention, Bayesian inversion isused for the 3D reconstruction. Input data for the algorithm is the setof projection images and the following a priori knowledge:

-   -   (a) Dental tissue consists of few approximately homogeneous        regions with well-defined boundaries.    -   (b) Dental tissue can only attenuate X-radiation, not intensify        it.

The example of the detector 6 used In the first preferred embodiment ofthe invention is based on charge coupled device (CCD) technology and hasdynamic range of 4096 gray levels. The size of the active imaging areais 34 mm*26 mm and the resolution is 872*664 pixels. After exposure,each pixel contains an integer which is proportional to the number ofX-ray quanta that hit the pixel's area.

Alternative detectors include any other digital intraoral sensor,digitized X-ray film, or any intraoral sensing device convertingdetected X-ray photons to a digital image.

In the measurement geometry the focal point of the X-ray source 2 moveshorizontally on a circle in the plane 10, with center at the detector 6,see FIG. 8. In FIG. 8, the detector 6 corresponds to one row of CCDpixels in the CCD sensor.

In FIG. 8 is presented an example of taking seven radiographs of someteeth of a patient. The total opening angle of the projection views is55 degrees in this example. FIG. 8 represents the xy plane, and thez-coordinate axis is thought to be perpendicular to the plane of thepaper.

In the preferred embodiments, mathematical modelling of the object 4 canbe done with a prior distribution of the form in formula (7):$\begin{matrix}{{p_{pr}(x)} = {\exp\left( {{- \alpha}{\sum\limits_{N}{U_{N}(x)}}} \right)}} & (7)\end{matrix}$where the sum is taken over a collection of 3D neighbourhoods N and thevalue U_(N)(x) depends only on the values of voxels belonging to theneighborhood N, and a is a positive regularization parameter used totune the width of the prior distribution. The neighborhood N istypically a set of voxels whose center points are doser to each otherthan some predefined maximum distance.

It is to be noted that formula (7) does not define a probabilitydistribution since it does not integrate to 1. Instead, the integraldiverges. However, when used In the Bayes formula (4) with a suitablelikelihood distribution, the posterior distribution is integrable. Thesame remark concerns the priors derived from (7), that is, formulae (9),(10) and (11).

In the preferred embodiments of the invention the three-dimensionalproblem can be reduced to a stack of two-dimensional problems eachcorresponding to a plane determined by a constant value of z. Here FIG.8 represents exactly the situation in the xy plane, i.e. the plane z=0.Each row in the detector corresponds to one such 2D problem. Thisapproach leads to small approximation error because the X-ray sourcedoes not move in the correct plane for nonzero z, and this error isneglected.

Next the modeling of a 2D tomographic problem is explained. Letm(j)=A×(j)+e(j)   (8)denote the jth 2D tomographic problem. Here the vector m(j) contains thereadings on jth row from each of the seven radiographs. The vector x(j)is the jth slice of the 3-D representation x of the object 4 underimaging. The matrix A is the pencil beam model matrix for the 2-Dtomography problem.

Now x(j) is a 2D array of pixels. The pixels are denoted by x(j)[k,q],where k=1,2, . . . ,K is row index and q=1,2, . . . ,Q is column index.

The mathematical modelling of the object 4, i.e. incorporation of priorinformation, is next explained for the 2D slice. Define $\begin{matrix}{{p_{pr}\left( x^{(j)} \right)} = {\exp\left( {{- \alpha}{\sum\limits_{k = 1}^{K}{\sum\limits_{q = 1}^{Q}{U\left( {N\left( {x^{(j)}\left\lbrack {k,q} \right\rbrack} \right)} \right)}}}} \right)}} & (9)\end{matrix}$where the value U(N(x(o)[k,q])) depends only on the values of pixelsbelonging to the neighborhood N of the pixel x(j)[k,q], the constant cis a normalization constant and a is a positive regularization parameterused to tune the width of the prior distribution. The choice of theneighborhoods N(x(j)[k,q]) is arbitrary, but a typical choice is the setof pixels whose centerpoints are doser to the centerpoint of x(j)[k,q]than some predefined distance.

In the preferred embodiment, the neighborhoods are chosen to consist oftwo adjacent pixels. Further, the function U is chosen to be a power ofabsolute value of the difference between adjacent pixels. These choiceslead to formula (10): $\begin{matrix}{{p_{pr}\left( x^{(j)} \right)} = {\exp\left( {- {\alpha\left( {\sum\limits_{k = 1}^{K - 1}{\sum\limits_{q = 1}^{Q}{{{{{x^{(j)}\left\lbrack {k,q} \right\rbrack} - {x^{(j)}\left\lbrack {{k + 1},q} \right\rbrack}}}^{s}++}{\sum\limits_{k = 1}^{K}{\sum\limits_{q = 1}^{Q - 1}{{{x^{(j)}\left\lbrack {k,q} \right\rbrack} - {x^{(j)}\left\lbrack {k,{q + 1}} \right\rbrack}}}^{s}}}}}} \right)}} \right)}} & (10)\end{matrix}$where s is a positive real number, and a is a regularization parameterused to tune the width of the prior distribution.

In the equation (11) is presented non-normalized total variation (TV)distribution, when s=1 in equation (10). This is the case according tothe preferred embodiment of the invention, because when s=1 the priormodel becomes such that it gives high probability for objects consistingof a few areas of different attenuation with well-defined boundaries.$\begin{matrix}{{{prTV}\left( x^{(j)} \right)} = {\exp\left( {- {\alpha\left( {\sum\limits_{k = 1}^{K - 1}{\sum\limits_{q = 1}^{Q}{{{{{x^{(j)}\left\lbrack {k,q} \right\rbrack} - {x^{(j)}\left\lbrack {{k + 1},q} \right\rbrack}}}++}{\sum\limits_{k = 1}^{K}{\sum\limits_{q = 1}^{Q - 1}{{{x^{(j)}\left\lbrack {k,q} \right\rbrack} - {x^{(j)}\left\lbrack {k,{q + 1}} \right\rbrack}}}}}}}} \right)}} \right)}} & (11)\end{matrix}$

X-radiation can only attenuate, not strengthen, inside tissue. Thisleads to positivity prior pos defined bypos(x(j))=1 if all pixels of x(j) are positive, 0 otherwise.   (12)

The explanation of modeling of one 2D tomographic problem is nowcomplete.

The stack of 2D problems are connected to each other by demanding thatconsecutive 2D slices x(j) and x(j-1) should not be very differentMathematically this Is expressed as follows:pr3D(x(j))=exp(−γΣΣ|x(j)[k,q]−x(j-1)[k,q]|),   (13)where the sums are taken over k=1, . . . ,K and q=1, . . . ,Q and γ>0 isanother regularization parameter. So the prior distribution in formula(4) is the product of (11), (12) and (13):p _(pr)(x(j))=prTV(x(j))pos(x(j))pr3D(x(j)).   (14)

The measurements are taken into account in the form of the likelihooddistribution $\begin{matrix}{{p\left( m^{(j)} \middle| x^{(j)} \right)} = {c\quad\exp\left\{ {{- \frac{1}{2}}\left( {m^{(j)} - {Ax}^{(j)}} \right)^{T}{\sum\limits^{- 1}\left( {m^{(j)} - {Ax}^{(j)}} \right)}} \right\}}} & (15)\end{matrix}$where Σ is the covariance matrix of the Gaussian noise vector e. c is anormalization constant.

In reality, X-radiation measurement noise is Poisson distributed.Formula 15 uses a Gaussian approximation to the Poisson distribution.

All the parts are presented of the right hand side of (4). Next it islooked for the MAP estimate of x(j), that is, the image x(j) that givesthe largest value for the posteriori distribution (4). This isequivalent to finding the image x(j) that minimizes the expression$\begin{matrix}{{F\left( x^{(j)} \right)} = {{\frac{1}{2}\left( {m^{(j)} - {Ax}^{(j)}} \right)^{T}{\sum\limits^{- 1}\left( {m^{(j)} - {Ax}^{(j)}} \right)}} + {- {\log\left( {{prTV}\left( x^{(j)} \right)} \right)}} - {\log\left( {{pr3D}\left( x^{(j)} \right)} \right)}}} & (16)\end{matrix}$with the additional requirement that every pixel of x(j) is positive.

Minimization of (16) is difficult since F Is not a differentiablefunction due to the absolute values present in (11) and (13) and due tothe sharp cutting in (12). The absolute values in (11) and (13) arereplaced by $\begin{matrix}{{{{t} \approx {h_{\beta}(t)}} = {\frac{1}{\beta}{\log\left( {\cosh\left( {\beta\quad t} \right)} \right)}}},} & (17)\end{matrix}$where β>0, enabling the use of efficient gradient-based minimizationroutines. The positivity constraint is taken care of by an interiorsearch minimization method. The results are shown in FIG. 9 incomparison with traditional tomosynthesis technique based onbackprojection.

In this description explained mathematic modelling and other imageinformation processing are performed by software means 15 to accomplishthree-dimensional information of the object 4.

In the second preferred embodiment the invention is utilized in dentalpanoramic x-ray imaging. The object 4 is typically teeth of the patientand the medical x-ray device 5 arrrangement is dental panoramic x-raydevice arrangement.

In the third preferred embodiment the invention is utilized Inmammography, for example in FFDM (Full Field Digital Mammography). Therethe object 4 is breast of human being and the medical x-ray device 5arrangement is mammography device arrangement.

In the fourth preferred embodiment the invention is utilized in surgicalx-ray imaging. There the object 4 is the body part under surgery and themedical x-ray device 5 arrangement is surgical x-ray device arrangement.

In the first, second, third and fourth preferred embodiments of theinvention the basic method steps are same as mentioned with the flowchart In FIG. 6. The utilizing of the invention In the second, third andfourth preferred embodiment is similar to what is described with thefirst preferred embodiment of the invention and elsewhere in thisapplication except different medical x-ray imaging applications andtheir differences because of different medical x-ray devices anddifferent objects to be x-ray imaged.

Although the invention is described above with reference to the examplesillustrated in the attached figures, it is to be understood that theinvention is not limited thereto but can be varied in many ways withinthe inventive idea disclosed in the attached claims.

1. A method for producing three-dimensional information of an object (4)in medical X-ray imaging, characterized in that the object is modelledmathematically independently of X-ray imaging, the object is X-radiatedfrom at least two different directions and the said X-radiation isdetected to form projection data of the object (4), and said projectiondata and said mathematical modelling of the object are utilized inBayesian inversion based on Bayes' formula${p\left( x \middle| m \right)} = \frac{{p_{pr}(x)}{p\left( m \middle| x \right)}}{p(m)}$to produce three-dimensional information of the object, the priordistribution p_(pr)(x) representing mathematical modelling of theobject, x representing the object image vector, which comprises valuesof the X-ray attenuation coefficient inside the object, m representingprojection data, the likelihood distribution p(m|x) representing theX-radiation attenuation model between the object Image vector x andprojection data m, p(m) being a normalization constant and theposteriori distribution p(x|m) representing the three-dimensionalinformation of the object (4).
 2. A method according to claim 1,characterized in that the three-dimensional information of the object(4) is one or more two-dimensional Images representing X-ray attenuationcoefficient along slices through the object.
 3. A method according toclaim 1, characterized in that the three-dimensional information of theobject (4) is a three-dimensional voxel representation of the X-rayattenuation in the object
 4. A method according to claim 1,characterized in that the measurement model is m=Ax+e, where matrix Acontains the lengths of the path of the X-ray inside each voxel and thenoise e is independent of object image vector x leading to thelikelihood distributionp(m|x)=p _(noise)(m−Ax).
 5. A method according to claim 1, characterizedin that the said mathematical modelling comprises that X-radiationattenuates when passing the object (4), which means that every imagevoxel is nonnegative.
 6. A method according to claim 1, characterized inthat mathematical modelling is expressed by the formula:${p_{pr}(x)} = {\exp\left( {{- \alpha}{\sum\limits_{N}{U_{N}(x)}}} \right)}$where the sum is taken over a collection of 3D neighbourhoods N and thevalue U_(N)(x) depends only on the values of voxels belonging to theneighborhood N, and α is a positive regularization parameter used totune the width of the prior distribution.
 7. A method according to claim1, characterized in that the 3D tomographic problem Is divided into astack of 2D tomographic problems and on every such 2D problem, themathematical modelling is expressed by the formula:${p_{pr}(x)} = {\exp\left( {{- \alpha}{\sum\limits_{N}{U_{N}(x)}}} \right)}$where the sum is taken over a collection of 2D neighbourhoods N and thevalue U_(N)(x) depends only on the values of pixels belonging to theneighborhood N, and α is a positive regularization parameter used t tunethe width of the prior distribution, and the 2D tomographic problems arerelated to each other by the formulapr3D(x(j))=exp(−γΣΣ|x(j)[k,q]−x(j-1)[k,q]|), where the sums are takenover all pixels (k=1, . . . ,K, q=1, . . . ,Q) and γ>0 is anotherregularization parameter.
 8. A method according to claim 7,characterized in that the neighborhoods consist of two adjacent pixelsand U calculates a power of the absolute value of the difference,leading to the formula${p_{pr}\left( x^{(j)} \right)} = {\exp\left( {{- \alpha}\left( {{\sum\limits_{k = 1}^{K - 1}{\sum\limits_{q = 1}^{Q}{{{x^{(j)}\left\lbrack {k,q} \right\rbrack} - {x^{(j)}\left\lbrack {{k + 1},q} \right\rbrack}}}^{s}}} + {\sum\limits_{k = 1}^{K}{\sum\limits_{q = 1}^{Q - 1}{{{x^{(j)}\left\lbrack {k,q} \right\rbrack} - {x^{(j)}\left\lbrack {k,{q + 1}} \right\rbrack}}}^{s}}}} \right)} \right)}$where s is a positive real number.
 9. A method according to claim 8,characterized in that s=1 corresponding to total variation (TV) priordescribing objects (4) consisting of different regions with well-definedboundaries.
 10. A method according to claim 1, characterized in thatmathematical modelling is qualitative structural information of thetarget where the structural information is encoded in priordistributions that are concentrated around object image vectors x thatcorrespond to the physiological structures of the object (4).
 11. Amethod according to claim 1, characterized in that mathematicalmodelling consists of a list or probability distribution of possibleattenuation coefficient values in the object (4).
 12. A method accordingto claim 1, characterized in that the X-ray imaging geometry, such asX-ray source position, has unknown error modelled in the distributionp(m|x).
 13. A method according to claim 1, characterized in that theX-radiation measurement noise is Poisson distributed.
 14. A methodaccording to claim 1, characterized in that the medical X-ray imaging isdental radiography.
 15. A method according to claim 1, characterized inthat the medical X-ray imaging is surgical C-arm imaging.
 16. A methodaccording to claim 1, characterized in that the medical X-ray imaging ismammography.
 17. A method according to claim 1, characterized in thatthree-dimensional information of the object (4) is produced on the basisof the maximum a posteriori estimator (MAP) which is calculated by theequation:p(xMAP|m)=maxp(x|m), m representing projection data and x representingthe object image vector and where the maximum on the right hand side ofthe equation is taken over all x.
 18. A method according to claim 1,characterized in that three-dimensional information of the object (4) isproduced on the basis of the conditional mean estimator (CM), which Iscalculated by the equation:x _(CM) =∫xp(x|m)dx where m represents projection data and x representsthe object image vector.
 19. A medical X-ray device (5) arrangement forproducing three-dimensional information of an object (4) in a medicalX-ray imaging, characterized in that the medical X-ray device (5)arrangement comprises: means (15) for modelling the object (4)mathematically independently of X-ray imaging an X-ray source (2) forX-radiating the object from at least two different directions a detector(6) for detecting the X-radiation to form projection data of the object(4) and means (15) for utilizing said projection data and saidmathematical modelling of the object in Bayesian inversion based onBayes' formula${p\left( x \middle| m \right)} = \frac{{p_{pr}(x)}{p\left( m \middle| x \right)}}{p(m)}$to produce three-dimensional information of the object, the priordistribution p_(pr)(x) representing mathematical modelling of theobject, x representing the object image vector, which comprises valuesof the X-ray attenuation coefficient inside the object, m representingprojection data, the likelihood distribution p(m|x) representing theX-radiation attenuation model between the object image vector x andprojection data m, p(m) being a normalization constant and theposteriori distribution p(x|m) representing the three-dimensionalinformation of the object (4).
 20. A medical x-ray device (5)arrangement according to claim 19, characterized in that thethree-dimensional information of the object (4) is one or moretwo-dimensional images representing X-ray attenuation coefficient alongslices through the object.
 21. A medical x-ray device (5) arrangementaccording to claim 19, characterized in that the three-dimensionalinformation of the object (4) is a three-dimensional voxelrepresentation of the X-ray attenuation in the object.
 22. A medicalX-ray device (5) arrangement according to claim 19, characterized inthat the medical X-ray device arrangement comprises means (15) formodelling the measurement asm=Ax+e, where matrix A contains the, lengths of the path of the X-rayinside each voxel and the noise e is independent of object image vectorx leading to the likelihood distributionp(m|x)=p _(noise)(m−Ax).
 23. A medical X-ray device (5) arrangementaccording to claim 19 characterized in that the medical X-ray devicearrangement comprises means (15) for modelling the object (4)mathematically so that X-radiation attenuates when passing the object(4), which means that every image voxel is nonnegative.
 24. A medicalX-ray device (5) arrangement according to claim 19, characterized inthat the medical X-ray device arrangement comprises means (15) formodelling the object (4) mathematically by the formula:${p_{pr}(x)} = {\exp\left( {{- \alpha}{\sum\limits_{N}{U_{N}(x)}}} \right)}$where the sum is taken over a collection of 3D neighbourhoods N and thevalue U_(N)(x) depends only on the values of voxels belonging to theneighborhood N, and α is a positive regularization parameter used totune the width of the prior distribution.
 25. A medical x-ray device (5)arrangement according to claim 19, characterized in that the 3Dtomographic problem is divided into a stack of 2D tomographic problemsand on every such 2D problem, and the medical X-ray device arrangementcomprises means (15) for modelling the object (4) mathematically by theformula:${p_{pr}(x)} = {\exp\left( {{- \alpha}{\sum\limits_{N}{U_{N}(x)}}} \right)}$where the sum is taken over a collection of 2D neighbourhoods N and thevalue U_(N)(x) depends only on the values of pixels belonging to theneighborhood N, and α is a positive regularization parameter used totune the width of the prior distribution, and the 2D tomographicproblems are related to each other by the formulapr3D(x(j))=exp(−γΣΣ|x(j)[k,q]−x(j-1)[k,q]|), where the sums are takenover all pixels (k=1, . . . ,K, q=1, . . . ,Q) and γ>0 is anotherregularization parameter.
 26. A medical X-ray device (5) arrangementaccording to claim 25, characterized in that the neighborhood systemsconsist of two neighboring pixels xj, xk or voxels xj, xk and U_(N)(x)calculates a power of the${p_{pr}\left( x^{(j)} \right)} = {\exp\left( {{- \alpha}\left( {{\sum\limits_{k = 1}^{K - 1}{\sum\limits_{q = 1}^{Q}{{{x^{(j)}\left\lbrack {k,q} \right\rbrack} - {x^{(j)}\left\lbrack {{k + 1},q} \right\rbrack}}}^{s}}} + {\sum\limits_{k = 1}^{K}{\sum\limits_{q = 1}^{Q - 1}{{{x^{(j)}\left\lbrack {k,q} \right\rbrack} - {x^{(j)}\left\lbrack {k,{q + 1}} \right\rbrack}}}^{s}}}} \right)} \right)}$absolute value of the difference, leading to the formula where s is apositive real number and α is a regularization parameter used to tunethe width of the prior distribution.
 27. A medical X-ray device (5)arrangement according to claim 26, characterized in that the medicalX-ray device arrangement comprises means (15) for modelling the object(4) mathematically by setting s=1 corresponding to total variation (TV)prior describing objects consisting of different regions withwell-defined boundaries.
 28. A medical X-ray device (5) arrangementaccording to claim 19, characterized in that the medical X-ray devicearrangement comprises means (15) for modelling the object (4)mathematically by assuming that mathematical modelling is qualitativestructural information of the target where the structural information isencoded In prior distributions that are concentrated around imagevectors x that correspond to the physiological structures of the target.29. A medical X-ray device (5) arrangement according to claim 19,characterized in that the medical X-ray device arrangement comprisesmeans (15) for modelling the object (4) mathematically by assuming thatmathematical modelling consists of a list of possible attenuationcoefficient values in the object.
 30. A medical X-ray device (5)arrangement according to claim 19, characterized in that the medicalX-ray device arrangement comprises means (15) for modelling the object(4) mathematically by assuming that the X-ray imaging geometry, such asX-ray source position, has unknown error modelled In the distributionp(m|x).
 31. A medical X-ray device (5) arrangement according to claim19, characterized in that the medical X-ray device arrangement comprisesmeans (15) for modelling the object (4) mathematically by assuming thatX-radiation measurement noise is Poisson distributed.
 32. A medicalX-ray device (5) arrangement according to claim 19, characterized inthat the medical X-ray imaging is dental radiography.
 33. A medicalX-ray device (5) arrangement according to claim 19, characterized inthat the medical X-ray imaging is surgical C-arm imaging.
 34. A medicalX-ray device (5) arrangement according to claim 19, characterized inthat the medical X-ray imaging is mammography.
 35. A medical X-raydevice (5) arrangement according to claim 19, characterized in that themedical X-ray device arrangement comprises means (15) for producingthree-dimensional information of the object (4) on the basis of themaximum a posteriori estimator (MAP), which is calculated by theequation:p(xMAP|m)=maxp(x|m), m representing projection data and x representingthe object image vector and where the maximum on the right hand side ofthe equation is taken over all x.
 36. A medical X-ray device (5)arrangement according to claim 19, characterized in that the medicalX-ray device arrangement comprises means (15) for producingthree-dimensional Information of the object (4) on the basis of theconditional mean estimator (CM), which is calculated by the equationx _(CM) =∫xp(x|m)dx where m represents projection data and x representsthe object image vector.